Found problems: 85335
1996 Tournament Of Towns, (486) 4
All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle.
(a) Draw a hexagon such that it has the largest possible number of points of self-intersection.
(b) Prove that this number is indeed maximum.
(NB Vassiliev)
2011 Princeton University Math Competition, A5
Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.
2019 Jozsef Wildt International Math Competition, W. 50
Let $x$, $y$, $z > 0$, $\lambda \in (-\infty, 0) \cup (1,+\infty)$ such that $x + y + z = 1$. Then$$\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}$$
2012 Stanford Mathematics Tournament, 3
Let $ABC$ be an equilateral triangle of side 1. Draw three circles $O_a$, $O_b$, $O_c$ with diameters $BC$, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of intersection between the three circles. Find $S_a+S_b+S_c-S$.
1992 Poland - First Round, 6
The sequence $(x_n)$ is determined by the conditions:
$x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$.
Find $\sum_{n=0}^{1992} 2^nx_n$.
2023 Brazil EGMO Team Selection Test, 3
Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.
2016 AMC 12/AHSME, 7
Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 56 \qquad
\textbf{(D)}\ 64 \qquad
\textbf{(E)}\ 96$
2024 Myanmar IMO Training, 4
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
2008 National Olympiad First Round, 36
There is a white table with a pile of $2008$ coins and there are two empty black tables. At each move, the uppermost coin on a table is transferred to an empty table or to the top of the pile on a non-empty table. What is the least number of moves required to reverse the pile at the beginning on the white table?
$
\textbf{(A)}\ 6016
\qquad\textbf{(B)}\ 6017
\qquad\textbf{(C)}\ 6022
\qquad\textbf{(D)}\ 6023
\qquad\textbf{(E)}\ 6024
$
1989 APMO, 2
Prove that the equation \[ 6(6a^2 + 3b^2 + c^2) = 5n^2 \] has no solutions in integers except $a = b = c = n = 0$.
2021 AMC 10 Spring, 11
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
$(\textbf{A}) \: 24 \qquad (\textbf{B}) \: 30 \qquad (\textbf{C}) \: 48 \qquad (\textbf{D}) \: 60 \qquad (\textbf{E}) \: 64$
2014 Online Math Open Problems, 18
Find the number of pairs $(m,n)$ of integers with $-2014\le m,n\le 2014$ such that $x^3+y^3 = m + 3nxy$ has infinitely many integer solutions $(x,y)$.
[i]Proposed by Victor Wang[/i]
1973 AMC 12/AHSME, 34
A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was
$ \textbf{(A)}\ 54 \text{ or } 18 \qquad
\textbf{(B)}\ 60 \text{ or } 15 \qquad
\textbf{(C)}\ 63 \text{ or } 12 \qquad
\textbf{(D)}\ 72 \text{ or } 36 \qquad
\textbf{(E)}\ 75 \text{ or } 20$
2022 Novosibirsk Oral Olympiad in Geometry, 3
In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon.
[img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]
2021 Yasinsky Geometry Olympiad, 3
In the triangle $ABC$, $h_a, h_b, h_c$ are the altitudes and $p$ is its half-perimeter. Compare $p^2$ with $h_ah_b + h_bh_c + h_ch_a$.
(Gregory Filippovsky)
2018 Brazil Undergrad MO, 23
How many prime numbers $ p $ the number $ p ^ 3-4 p + 9 $ is a perfect square
1985 Spain Mathematical Olympiad, 6
Let $OX$ and $OY$ be non-collinear rays. Through a point $A$ on $OX$, draw two lines $r_1$ and $r_2$ that are antiparallel with respect to $\angle XOY$. Let $r_1$ cut $OY$ at $M$ and $r_2$ cut $OY$ at $N$. (Thus, $\angle OAM = \angle ONA$). The bisectors of $ \angle AMY$ and $\angle ANY$ meet at $P$. Determine the location of $P$.
2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
PEN P Problems, 34
If $n$ is a positive integer which can be expressed in the form $n=a^{2}+b^{2}+c^{2}$, where $a, b, c$ are positive integers, prove that for each positive integer $k$, $n^{2k}$ can be expressed in the form $A^2 +B^2 +C^2$, where $A, B, C$ are positive integers.
2019 Latvia Baltic Way TST, 3
All integers are written on an axis in an increasing order. A grasshopper starts its journey at $x=0$. During each jump, the grasshopper can jump either to the right or the left, and additionally the length of its $n$-th jump is exactly $n^2$ units long. Prove that the grasshopper can reach any integer from its initial position.
2023 Indonesia MO, 5
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.
2019 Nepal TST, P2
Let $H$ be orthocenter of an acute $\Delta ABC$, $M$ is a midpoint of $AC$. Line $MH$ meets lines $AB, BC$ at points $A_1, C_1$ respectively, $A_2$ and $C_2$ are projections of $A_1, C_1$ onto line $BH$ respectively. Prove that lines $CA_2, AC_2$ meet at circumscribed circle of $\Delta ABC$.
[i]Proposed by Anton Trygub[/i]
1965 Vietnam National Olympiad, 1
At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$.
1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish?
2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?
1961 AMC 12/AHSME, 10
Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is:
${{ \textbf{(A)}\ \sqrt{18} \qquad\textbf{(B)}\ \sqrt{28} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \sqrt{63} }\qquad\textbf{(E)}\ \sqrt{98} } $
MathLinks Contest 2nd, 4.2
Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.